The processing of local action will not by default progress the flow to the next assignment and will remain on the same assignment and the routing will still work as defined on the assignment irrespective of the local action.
Is it your requirement here to progress the case to the next assignment in the flow on processing of the local action? That will need some customisation.
Already User picked the Assignment from WB. Once picked user is not performing the flow action instead he switch to Local action and once local action submitted.Flow is again routed as routing logic is there on assignment.
Here my case routing logic need to be there on assignment itself but when user submit the local action it should not get routed again.